25. Independent But Not Identically Distributed Random Variables

0. INTRODUCTION We now return to the situation of Section 3.2 in which are independent with df's . Good theorems for the empirical process in this present situation will require an extension of the DKW inequality (Inequality 9.2.1). This is the subject of Section 1. Just as , in the present situation the marginal of the empirical process at a fixed point has the generalized binomial distribution. In a very strong sense the generalized binomial distribution is less dispersed than an appropriately chosen ordinary binomial distribution. This fact is the subject of Section 2. This result is then used to obtain in probability “linear bounds” on the empirical df in Section 3. Armed with these inequalities, we then establish the weak convergence of the empirical, weighted empirical, and quantile processes of independent but not identically distributed rv's in ‖ /q‖ metrics in Section 4. Section 5 extends our earlier work on L-statistics to this case.